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    On the adjacent vertex distinguishing edge chromatic number of graphs.
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    Keywords:
    Brooks' theorem
    A proper edge coloring of graph G is called equitable adjacent strong edge coloring if colored sets from every two adjacent vertices incident edge are different, and the number of edges in any two color classes differ by at most one, which the required minimum number of colors is called the adjacent strong equitable edge chromatic number. In this paper, we present the edge coloring of join-graphs about path and cycle, and gain the vertex-distinguishing edge chromatic number of 𝑃𝑚 ∨ 𝐶𝑛.
    Edge Coloring
    Brooks' theorem
    Fractional coloring
    Graph Coloring
    Coloring problem is a classical difficult problem of graph theory. It is a fundamental problem in scientific computation and engineering design. In recent years, a variety of graph coloring problems frequently appeared and solved many problems in production. It is a difficult problem to discuss the chromatic number of a given graph class. In the paper, we introduce several kinds of chromatic numbers of graphs such as adjacent-vertex-distinguishing total chromatic number, adjacent-vertex-distinguishing proper edge chromatic number, smarandachely-adjacent-vertex-distinguishing edge chromatic number, and the multi-fan graphs are considered.
    Windmill graph
    Graph Coloring
    Brooks' theorem
    Edge Coloring
    It was proved that for any integer k≥2,there was a k+1-regular graph with vertex-distinguishing edge chromatic number 2k+1,and for any integer m≥4,there was a bipartite graph with vertex-distinguishing edge chromatic number m.
    Windmill graph
    Edge Coloring
    Citations (0)
    The vertex-distinguishing edge-coloring of F_m∨P_n are studied and vertex-distinguishing edge chromatic number of F_m∨P_n are obtained.
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    Let G be a simple connected graph with ordr not less than 3,k-proper edge coloring of G is called adjacent-vertex distinguishing,if for arbitrary two adjacent vertices which are incident to different sets of colored edges.The minimum number required for an adjacent-vertex distinguishing edge coloring(AVDEC) of G is called the adjacent strong edge chromatic number.In this paper,we give two upper bounds of adjacent-vertex distinguishing edge chromatic number of Cartesian product graphs,and some results are obtained.
    Cartesian product
    Edge Coloring
    Neighbourhood (mathematics)
    Brooks' theorem
    Simple graph
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    The adjacent vertex-distinguishing edge chromatic number of join graph of circle and complete graph is obtained in this paper.
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